..., λ) , x λ + y ν = (x + y) min (λ, v) , d µ o x λ + c K o y v = (dx + cy) min{µ , λ, k, v}. Proof: Left for the reader to refer [75]. The... ...y for all x ∈ V, (K o A)(x) = sup{min {K(c), A(y)}c ∈ P ⊂ R , y ∈ V, x = cy P is a subfield in R relative to which V is defined}. DEFINITION 4.... .... x λ + y ν = (x + y) min (λ, ν) , d µ o x λ + c κ o y ν = (dx + cy) min{µ , λ, κ, ν}. Problem 4.5.59: Show if M is a δ-Noetherian module... ...r all x ∈ H ⊂ G (K o A)(x) = sup{min {K(c), A(y)}  c ∈ S, y ∈ H ⊂ G, x = cy}. Fuzzy singletons are defined as in case of any other fuzzy subset. ... ...,λ} , x λ + y ν = (x + y) min (λ ,ν) , dµ o x λ + c R o y ν = (dx + cy) min (µ, λ, k, ν} . Problem 5.5.39: Prove the fuzzy subset ν of SF (... ...49. JACOBSON, Nathan, Basic Algebra I, Freeman, New York, 1974. 50. JUN, Young Bae, NEGGERS, J. and KIM, Hee Sik, Normal L-fuzzy ideals in semiring... ...c groups, J. Fuzzy Math., 3, 465-470 (1995). 56. KIM, Kyung Ho, and JUN, Young Bae, Normal fuzzy R-subgroups in near- rings, Fuzzy Sets and Systems,... ...ximations in a fuzzy group, Inform. Sci., 90, 203-220 (1991). 71. KWON, Young-In, Fuzzy Near-ring modules over fuzzy near-rings, Pusan Kyongnam Ma...

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